An integral equation approach to kinematic dynamo models

The paper deals with dynamo models in which the induction effects act withi n a bounded region surrounded by an electrically conducting medium at rest. Instead of the induction equation, an equivalent integral equation is cons idered, which again poses an eigenvalue problem. The eigenfunctions and eig envalues represent the magnetic field modes and corresponding dynamo number s. In the simplest case, that is for homogeneous conductivity and steady fi elds, this integral equation follows immediately from the Biot-Savart law. For this case, numerical results are presented for some spherical and ellip tical axisymmetric alpha(2)omega-dynamo models. For a large class of models the interesting feature of dipole-quadrupole is found. Using Green's funct ion of a Helmholtz-type equation, we derive a more general integral equatio n, which applies to time-dependent magnetic field modes, too, and gives us some insight into the spectral properties of the integral operators involve d. In particular, for homogeneous conductivity the operator is compact and thus bounded, which leads to a necessary condition for dynamo action.